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We develop methods for the analysis of a collection of curves which are stochastically modelled as independent realizations of a random function with an unknown mean and covariance structure. We propose a method of estimating the mean function nonparametrically under the assumption that it is smooth. We suggest a variant on the usual form of cross-validation for choosing the degree of smoothing to be employed. This method of cross-validation, which consists of deleting entire sample curves, has the advantage that it does not require that the covariance structure be known or estimated. In the estimation of the covariance structure, we are primarily concerned with models in which the first few eigenfunctions are smooth and the eigenvalues decay rapidly, so that the variability is predominantly of large scale. We propose smooth non parametric estimates of the eigenfunctions and a suitable method of cross-validation to determine the amount of smoothing. Our methods are applied to data on the gaits of a group of 5-year-old children.

SUMMARY Non‐parametric regression using cubic splines is an attractive, flexible and widely‐applicable approach to curve estimation. Although the basic idea was formulated many years ago, the method is not as widely known or adopted as perhaps it should be. The topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology. Particular subjects covered include the basic principles of the method; the relation with moving average and other smoothing methods; the automatic choice of the amount of smoothing; and the use of residuals for diagnostic checking and model adaptation. The question of providing inference regions for curves – and for relevant properties of curves – is approached via a finite‐dimensional Bayesian formulation.

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coef®cients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior speci®ed, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any speci®c Besov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coef®cients. However, prior knowledge about a function's regularity properties might be dif®cult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.

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